Reduced-order model for the BGK equation based on POD and optimal transport

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Reduced-order model for the BGK equation based on POD and optimal transport

Florian Bernard, Angelo Iollo, Sébastien Riffaud

To cite this version:

Florian Bernard, Angelo Iollo, Sébastien Riffaud. Reduced-order model for the BGK equation based on POD and optimal transport. MOREPAS 2018 - Model Reduction of Parametrized Systems IV, Apr 2018, Nantes, France. �hal-02427339�

Reduced-order model for the BGK equation based on POD and optimal transport

S. Riffaud ^1,2 , F. Bernard ^1,2 , A. Iollo ^1,2

1

Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33400 Talence, France

2

Memphis team, Inria Bordeaux Sud-Ouest, 33400 Talence, France

Contribution

We present a model describing a gas flow in both hydrodynamic and rarefied regimes. Since the computational cost of the Bhatnagar- Gross-Krook (BGK) equation can be prohibitive, a reduced-order ap- proximation is developed, leading to fast and accurate simulations.

BGK equation

The dynamics of the gas flow is described by the BGK equation:

∂f

∂t (x, ξ, t) + ξ · ∇

_x

f (x, ξ, t) = M

_f

(x, ξ, t) − f (x, ξ, t)

τ (x, t) (1)

where f is the density distribution function representing the density of gas particles at point x ∈ R

³

, velocity ξ ∈ R

³

and time t ∈ R . The Maxwellian distribution function M

_f

is in dimensionless form

M

_f

(x, ξ, t) = ρ(x, t)

(2πT (x, t))

³²

exp

− kξ − U (x, t)k

²

2T (x, t)

where ρ ∈ R is the density, U ∈ R

³

is the macroscopic velocity, T ∈ R is the temperature and E ∈ R is the total energy of the gas.

Reduced-order approximation

The distribution functions are approximated by f e (x, ξ, t) =

N_pod

X

n=1

a

^f_n

(x, t)Φ

_n

(ξ )

and

M f

_f

(x, ξ, t) =

N_pod

X

n=1

a

^M_n

(x, t)Φ

_n

(ξ )

where the basis functions Φ

_n

are built offline by Proper Orthog- onal Decomposition (POD) and the coefficients a

_n

are computed online by the Galerkin method.

References

[1] P. L. Bhatnagar, E. P. Gross and M. Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Physical Review, 1954.

[2] F. Bernard. Efficient asymptotic preserving schemes for BGK and ES-BGK models on cartesian grids. PhD thesis, 2015.

[3] F. Bernard, A. Iollo and S. Riffaud. Reduced-order model for the BGK equation based on POD and optimal transport. Journal of Computational Physics, 2018.

Offline phase

In the offline phase, the BGK equation (1) is sampled to collect infor- mation on the distribution functions that we want to approximate.

High-fidelity simulations provide snapshots of both the density dis- tribution function and of the Maxwellian distribution function:

S

^hf

= n

f (x

_i

, ξ, t

_k

) o

1≤i≤N_x 1≤k≤N_t

[ n M

_f

(x

_i

, ξ, t

_k

) o

1≤i≤N_x 1≤k≤N_t

Then, optimal transport provides addi- tional low-fidelity snapshots by interpolating the snapshots of S

^hf

to complete the sampling:

S = S

^hf

∪ S

^lf

Finally, the basis functions Φ

_n

are built by POD to have the best approximation in the least squares sense of the snapshots s

_l

∈ S :



 

 

 

 

minimize

Φ₁(ξ)...Φ_Npod (ξ)

N_snaps

X

l=1

Z

R³

s

_l

(ξ ) − P [s

_l

](ξ )

2

dξ

subject to

Z

R³

Φ

_n

(ξ )Φ

_m

(ξ ) dξ = δ

_n,m

where P [s

_l

] is the projection of s

_l

onto the subspace spanned by the basis functions Φ

_n

i.e. P [s

_l

](ξ ) =

N_pod

P

n=1

R

R³

s

_l

(ξ

⁰

)Φ

_n

(ξ

⁰

) dξ

⁰

Φ

_n

(ξ ) .

Online phase

During the online phase, the offline knowledge is used to compute approximations of the distribution functions at low cost. In the Galerkin method, the BGK equation (1) is projected onto the basis functions Φ

_n

, leading to an hyperbolic system of partial differen- tial equations:

∂a

^f

∂t (x, t)+A ∂a

^f

∂x (x, t)+ ˚ A ∂a

^f

∂y (x, t)+

?

A ∂a

^f

∂z (x, t) = a

^M

(x, t) − a

^f

(x, t) τ (x, t)

where a = (a

₁

, a

₂

, . . . , a

_Npod

)

^T

, A

_n,m

= R

R³

ξ

_u

Φ

_n

(ξ )Φ

_m

(ξ ) dξ , A ˚

_n,m

= R

R³

ξ

_v

Φ

_n

(ξ )Φ

_m

(ξ ) dξ and A

^?_n,m

= R

R³

ξ

_w

Φ

_n

(ξ )Φ

_m

(ξ ) dξ . These equations are decoupled by linear changes of variables and are solved by an IMEX Runge-Kutta scheme in time and a finite volume scheme in space. To improve the accuracy of the model, the coefficients a

^M_n

are computed by constrained projection:



 

 

 

 

 

 

minimize

a^M₁ (x,t)...a^M

Npod (x,t)

Z

R³

M

_f

(x, ξ, t) − M f

_f

(x, ξ, t)

²

dξ

subject to

Z

R³

M f

_f

(x, ξ, t)





1 ξ

kξk² 2



 dξ =





ρ(x, t)

ρ(x, t)U (x, t) E (x, t)





in order to conserve mass, momentum and total energy of the gas.

Results

Kn = 0.0345 Kn = 0.0689

Kn = 0.115 Kn = 0.23

In average with N

_pod

= 20 basis functions, the approximation

error is less than 1% and the run time is divided by ap-

proximately 45 with respect to the high-fidelity simulations.